Fourier transformation, in particular discrete or fast Fourier transformation, is often used for the analysis of signals with regard to their signal components. In this case, it is generally endeavored to be able to determine as accurately as possible the frequencies and amplitudes of the individual signal components of the signal to be analyzed.
This can become problematic, however, if the signals are not captured regularly enough. In such cases, recourse is had for instance to various alias corrections, in order nevertheless to be able to carry out a reliable frequency analysis. Such a method is described for instance in US 2009/0231956 A1.
When determining the frequency there is generally a limitation on account of the so-called Nyquist-Shannon sampling theorem. In accordance therewith, a signal band-limited to a maximum frequency, in order that said signal can be reconstructed exactly again from the time-discrete signal, must be sampled at a sampling frequency greater than twice the bandwidth.
In practice, all systems are restricted with regard to the possible frequency resolution. The latter can indeed be improved by longer acquisition times or by apparently longer acquisition times, for which a signal is extended with “zeros”, this also being known by the designation zero padding. However, limits are imposed on the improvement, since the latter is generally associated with an increased computational complexity and/or longer measurement times.
A further problem is that, on account of the so-called leakage effect, the amplitude and/or frequency of a sinusoidal signal component contained in a signal in many cases cannot be reproduced correctly. This is attributable to the fact that there is always a finite observation period and the signal in reality fundamentally has a start time and an end time, which has the effect that frequency components which could not be found in the case of a non-realizable observation period of infinite length occur in a spectrum determined via Fourier analysis.
The leakage effect has the consequence that the amplitude and frequency of sinusoidal signal components are reproduced correctly only if the frequency lies exactly on one of the frequency lines of the spectrum. A widening of the peaks representing the signal components and a reduction of the amplitude occur in all other cases.
The use of different window functions is intended to reduce the leakage effect and to increase the amplitude accuracy for all frequency components. Window functions are functions which have the value zero outside a predefined interval. A signal to be analyzed is multiplied by the window function, such that it likewise assumes the value zero outside the predefined interval. The Fourier transformation is subsequently carried out.
In order to improve the frequency and amplitude accuracy, various procedures are known from the prior art. By way of example, peak fitting by means of parabolic, Gaussian or Lorentz curves is carried out.
Peak fitting using such curves is unsatisfactory here in some instances in practice, since the optimum curve shape is dependent on the window function used for the calculation of the spectra. Furthermore, there are a number of window functions, for example the so-called Tukey window, for which possibly no optimum curve shape can be found for fitting.
A further problem can arise as a result of noise contained in the signal or as a result of signal mixtures. By means of peak fitting, the frequency and/or amplitude of a sinusoidal component contained in the signal can then be determined only comparatively inaccurately.